Problem: Determine how many solutions exist for the system of equations. ${-8x+2y = 8}$ ${y = 6+2x}$
Solution: Convert both equations to slope-intercept form: ${-8x+2y = 8}$ $-8x{+8x} + 2y = 8{+8x}$ $2y = 8+8x$ $y = 4+4x$ ${y = 4x+4}$ ${y = 6+2x}$ ${y = 2x+6}$ Just by looking at both equations in slope-intercept form, what can you determine? ${y = 4x+4}$ ${y = 2x+6}$ The linear equations have different slopes. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ When two equations have different slopes, the lines will intersect once with one solution.